Time Optimal Swing-Up of the Planar Pendulum

This paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart. This mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open-loop controls. Relying on the theory of optimal synthesis, we provide a discontinuous feedback giving optimal solutions for any initial data. The approach is that of geometric control theory.